By S. I. Gelfand, M. L. Gerver, A. A. Kirillov, N. N. Konstantinov, A. G. Kushnirenko
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25), which will illustrate the method for proving the theorem in all cases. The idea is to write A as a sum of two terms and use the sum rule to separate off the term with a nonzero diagonal entry. 23) we write A W~(o S(X)H°0 fi X) A 0 )= M + A2(X). 30), the result is proved. • To study "reduction of order," let A be a 2 X 2 matrix continuous on [a,b] and consider the equation Y'(x) = A(x)Y(x). 32) We also assume, for illustrative purposes, that zl(x)¥=0 on [a,b]. 31), linearly independent from Y(x).
28), we obtain for any x,yB[a,b] the formula detft eA(s)* = e£*A{s)*m (1 30) rr y s In the ordinary theory of integration one proves the following additive property of the integral: fXf(s)ds= (Xf(s)ds+[yf(s)ds. 5. Let A:[a,b]^CnXn Then be continuous, and let x,j>,z€E[a,Z>]. f[eA(s)ds=f^eA(s)ds^eA(s)ds z y ( 1 3 2 ) z Proof. First suppose that z The same is true for the derivative y'. 6 Improper Product Integration In the usual theory of Riemann integration, one defines "improper" integrals in two types of situations. In the first type, the integration region is a finite interval [a, b] but there is difficulty at an endpoint. For instance, the function/to be integrated might be Riemann integrable over [a',b] for each af>a, but not Riemann integrable over [a,b]9 as in the integral f ——dx=\im[ l/2 •'O X e-*0Je —rjzdx. 1) l/2 X In the second type, the integration region is infinite, as in the integral o g Z 2 /•» 1 rM 1 I — dx = lim I — dx.